A unified approach to approximation algorithms for bottleneck problems
Journal of the ACM (JACM)
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Design networks with bounded pairwise distance
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Graph classes: a survey
Bounded-Diameter Minimum-Cost Graph Problems
Theory of Computing Systems
Decreasing the diameter of bounded degree graphs
Journal of Graph Theory
Decreasing the diameter of cycles
Journal of Graph Theory
Minimizing Average Shortest Path Distances via Shortcut Edge Addition
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Approximation algorithms for combinatorial problems
Journal of Computer and System Sciences
Minimizing the diameter of a network using shortcut edges
SWAT'10 Proceedings of the 12th Scandinavian conference on Algorithm Theory
Augmenting forests to meet odd diameter requirements
Discrete Optimization
Operations Research Letters
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In this paper, we study two variants of the problem of adding edges to a graph so as to reduce the resulting diameter. More precisely, given a graph G=(V,E), and two positive integers D and B, the Minimum-Cardinality Bounded-Diameter Edge Addition (MCBD) problem is to find a minimum-cardinality set F of edges to be added to G in such a way that the diameter of G+F is less than or equal to D, while the Bounded-Cardinality Minimum-Diameter Edge Addition (BCMD) problem is to find a set F of B edges to be added to G in such a way that the diameter of G+F is minimized. Both problems are well known to be NP-hard, as well as approximable within O(lognlogD) and 4 (up to an additive term of 2), respectively. In this paper, we improve these long-standing approximation ratios to O(logn) and to 2 (up to an additive term of 2), respectively. As a consequence, we close, in an asymptotic sense, the gap on the approximability of MCBD, which was known to be not approximable within clogn, for some constant c0, unless P=NP. Remarkably, as we further show in the paper, our approximation ratio remains asymptotically tight even if we allow for a solution whose diameter is optimal up to a multiplicative factor approaching 53. On the other hand, on the positive side, we show that at most twice of the minimal number of additional edges suffices to get at most twice of the required diameter. Some of our results extend to the edge-weighted version of the problems.